15.
设 P ( x ) = 24 x 24 + ∑ j = 1 23 ( 24 − j ) ( x 24 − j + x 24 + j ) . \begin{aligned} &P(x) = 24x^{24} \\ &\quad {}+ \sum_{j=1}^{23} (24 - j)\left(x^{24-j} + x^{24+j}\right). \end{aligned} P ( x ) = 24 x 24 + j = 1 ∑ 23 ( 24 − j ) ( x 24 − j + x 24 + j ) . z 1 , z 2 , … , z r z_1, z_2, \ldots, z_r z 1 , z 2 , … , z r P ( x ) P(x) P ( x ) k = 1 , 2 , … , r k = 1, 2, \ldots, r k = 1 , 2 , … , r z k 2 = a k + b k i z_k^2 = a_k + b_k i z k 2 = a k + b k i i = − 1 i = \sqrt{-1} i = − 1 a k a_k a k b k b_k b k ∑ k = 1 r ∣ b k ∣ = m + n p , \sum_{k=1}^{r} |b_k| = m + n\sqrt{p}, k = 1 ∑ r ∣ b k ∣ = m + n p , m m m n n n p p p p p p m + n + p m + n + p m + n + p
Let P ( x ) = 24 x 24 + ∑ j = 1 23 ( 24 − j ) ( x 24 − j + x 24 + j ) . \begin{aligned} &P(x) = 24x^{24} \\ &\quad {}+ \sum_{j=1}^{23} (24 - j)\left(x^{24-j} + x^{24+j}\right). \end{aligned} P ( x ) = 24 x 24 + j = 1 ∑ 23 ( 24 − j ) ( x 24 − j + x 24 + j ) . z 1 , z 2 , … , z r z_1, z_2, \ldots, z_r z 1 , z 2 , … , z r P ( x ) , P(x), P ( x ) , z k 2 = a k + b k i z_k^2 = a_k + b_k i z k 2 = a k + b k i k = 1 , 2 , … , r , k = 1, 2, \ldots, r, k = 1 , 2 , … , r , i = − 1 , i = \sqrt{-1}, i = − 1 , a k a_k a k b k b_k b k ∑ k = 1 r ∣ b k ∣ = m + n p , \sum_{k=1}^{r} |b_k| = m + n\sqrt{p}, k = 1 ∑ r ∣ b k ∣ = m + n p , m , m, m , n , n, n , p p p p p p m + n + p . m + n + p. m + n + p .
答案：15 解答： P ( x ) P(x) P ( x ) x k x^k x k 1 ≤ k ≤ 47 1 \le k \le 47 1 ≤ k ≤ 47 24 − ∣ 24 − k ∣ 24 - |24 - k| 24 − ∣24 − k ∣ x 24 x^{24} x 24 + 1 +1 + 1 − 1 -1 − 1 1 − x 1 - x 1 − x ( 1 − x ) P ( x ) = ( x + x 2 + ⋯ + x 24 ) − ( x 25 + ⋯ + x 48 ) = ( x + x 2 + ⋯ + x 24 ) ⋅ ( 1 − x 24 ) , \begin{aligned} &(1 - x)P(x) \\ &= (x + x^2 + \cdots + x^{24}) \\ &\quad {}- (x^{25} + \cdots + x^{48}) \\ &= (x + x^2 + \cdots + x^{24}) \\ &\quad {}\cdot (1 - x^{24}), \end{aligned} ( 1 − x ) P ( x ) = ( x + x 2 + ⋯ + x 24 ) − ( x 25 + ⋯ + x 48 ) = ( x + x 2 + ⋯ + x 24 ) ⋅ ( 1 − x 24 ) , x ≠ 1 x \ne 1 x = 1 P ( x ) = x ( x 24 − 1 x − 1 ) 2 . P(x) = x\left(\frac{x^{24} - 1}{x - 1}\right)^2. P ( x ) = x ( x − 1 x 24 − 1 ) 2 .
因此 P P P 0 0 0 1 1 1 24 24 24 z k = cos 15 k ∘ + i sin 15 k ∘ z_k = \cos 15k^\circ + i \sin 15k^\circ z k = cos 15 k ∘ + i sin 15 k ∘ k = 1 , … , 23 k = 1, \ldots, 23 k = 1 , … , 23 0 0 0 z k 2 = cos 30 k ∘ + i sin 30 k ∘ z_k^2 = \cos 30k^\circ + i \sin 30k^\circ z k 2 = cos 30 k ∘ + i sin 30 k ∘ ∣ b k ∣ = ∣ sin 30 k ∘ ∣ |b_k| = |\sin 30k^\circ| ∣ b k ∣ = ∣ sin 30 k ∘ ∣
当 k k k 1 1 1 12 12 12 ∣ sin 30 k ∘ ∣ |\sin 30k^\circ| ∣ sin 30 k ∘ ∣ 1 2 , 3 2 , 1 , 3 2 , 1 2 , 0 \frac{1}{2}, \frac{\sqrt{3}}{2}, 1, \frac{\sqrt{3}}{2}, \frac{1}{2}, 0 2 1 , 2 3 , 1 , 2 3 , 2 1 , 0 4 + 2 3 4 + 2\sqrt{3} 4 + 2 3 k = 13 , … , 23 k = 13, \ldots, 23 k = 13 , … , 23 k = 1 , … , 11 k = 1, \ldots, 11 k = 1 , … , 11 4 + 2 3 4 + 2\sqrt{3} 4 + 2 3 8 + 4 3 8 + 4\sqrt{3} 8 + 4 3 m + n + p = 8 + 4 + 3 = 15 m + n + p = 8 + 4 + 3 = 15 m + n + p = 8 + 4 + 3 = 15
The coefficient of x k x^k x k P ( x ) P(x) P ( x ) 24 − ∣ 24 − k ∣ 24 - |24 - k| 24 − ∣24 − k ∣ 1 ≤ k ≤ 47 , 1 \le k \le 47, 1 ≤ k ≤ 47 , + 1 +1 + 1 x 24 x^{24} x 24 − 1 -1 − 1 1 − x 1 - x 1 − x ( 1 − x ) P ( x ) = ( x + x 2 + ⋯ + x 24 ) − ( x 25 + ⋯ + x 48 ) = ( x + x 2 + ⋯ + x 24 ) ⋅ ( 1 − x 24 ) , \begin{aligned} &(1 - x)P(x) \\ &= (x + x^2 + \cdots + x^{24}) \\ &\quad {}- (x^{25} + \cdots + x^{48}) \\ &= (x + x^2 + \cdots + x^{24}) \\ &\quad {}\cdot (1 - x^{24}), \end{aligned} ( 1 − x ) P ( x ) = ( x + x 2 + ⋯ + x 24 ) − ( x 25 + ⋯ + x 48 ) = ( x + x 2 + ⋯ + x 24 ) ⋅ ( 1 − x 24 ) , x ≠ 1 , x \ne 1, x = 1 , P ( x ) = x ( x 24 − 1 x − 1 ) 2 . P(x) = x\left(\frac{x^{24} - 1}{x - 1}\right)^2. P ( x ) = x ( x − 1 x 24 − 1 ) 2 .
The distinct zeros of P P P 0 0 0 24 24 24 1 : 1: 1 : z k = cos 15 k ∘ + i sin 15 k ∘ z_k = \cos 15k^\circ + i \sin 15k^\circ z k = cos 15 k ∘ + i sin 15 k ∘ k = 1 , … , 23. k = 1, \ldots, 23. k = 1 , … , 23. 0 0 0 z k 2 = cos 30 k ∘ + i sin 30 k ∘ , z_k^2 = \cos 30k^\circ + i \sin 30k^\circ, z k 2 = cos 30 k ∘ + i sin 30 k ∘ , ∣ b k ∣ = ∣ sin 30 k ∘ ∣ . |b_k| = |\sin 30k^\circ|. ∣ b k ∣ = ∣ sin 30 k ∘ ∣.
As k k k 1 1 1 12 , 12, 12 , ∣ sin 30 k ∘ ∣ |\sin 30k^\circ| ∣ sin 30 k ∘ ∣ 1 2 , 3 2 , 1 , 3 2 , 1 2 , 0 \frac{1}{2}, \frac{\sqrt{3}}{2}, 1, \frac{\sqrt{3}}{2}, \frac{1}{2}, 0 2 1 , 2 3 , 1 , 2 3 , 2 1 , 0 4 + 2 3 ; 4 + 2\sqrt{3}; 4 + 2 3 ; k = 13 , … , 23 k = 13, \ldots, 23 k = 13 , … , 23 k = 1 , … , 11 k = 1, \ldots, 11 k = 1 , … , 11 4 + 2 3 . 4 + 2\sqrt{3}. 4 + 2 3 . 8 + 4 3 , 8 + 4\sqrt{3}, 8 + 4 3 , m + n + p = 8 + 4 + 3 = 15. m + n + p = 8 + 4 + 3 = 15. m + n + p = 8 + 4 + 3 = 15.
